1. Field of the Invention
The invention generally relates to the field of optical pulse characterization and, more particularly, to characterizing ultrashort optical pulses using simplified chronocyclic tomography.
2. Description of the Related Art
Traffic growth and the consequent demand for larger capacity optical data transmission systems have historically driven the increase of the per channel data rate of such systems to reduce the transmission cost per bit. Currently, 40 Gb/s transmission systems are commercially available and the feasibility of transmitting at much higher bit rates has been demonstrated.
Measuring and controlling the electric field of optical pulses is essential for ultra high bit rate transmission (e.g. 40 Gb/s+) due to the increased sensitivity to chromatic dispersion and optical nonlinearities. Measuring and controlling the electric field of optical pulses also facilitates the optimization of functions such as all-optical signal processing and nonlinear optical pulse compression.
Conventional optoelectronic photodetection and sampling are currently not capable of providing the time resolution and the phase sensitivity required to properly characterize pulses in an ultra high bit rate telecommunication environment. An effective diagnostic for optical sources in these environments requires time resolution better than 1-ps, and sub-mW sensitivity due to the low peak optical power of pulses used in telecommunication systems. Various characterization techniques have been proposed that rely on a nonlinear interaction, for example sum-frequency generation in a nonlinear crystal. These techniques however, lack the sensitivity required to operate in the telecommunication environment. Further, these techniques usually require an optical delay line, which strongly limits the measurement time of the experimental trace, therefore limiting the update rate of the measured electric field. Finally, some of these techniques are based on an iterative inversion of their experimental trace, and are therefore prone to errors or stagnation of the retrieval algorithm. There is therefore a need for techniques and experimental implementations providing reliable high sensitivity real-time measurement of the electric field of pulses in the telecommunication environment.
A simplified chronocyclic tomography technique has been proposed which allows the direct reconstruction of the electric field of a pulse from only two projections of its Wigner-Ville function. This technique is based on an analytic relation between the spectral intensity of a pulse I(ω), its spectral phase φ(ω), and the angular derivative of the frequency marginal of its rotated Wigner-Ville function. A rotation in the chronocyclic space generally requires quadratic spectral and temporal phase modulations, however, only temporal phase modulations are needed for simplified chronocyclic tomography. Specifically, the quadratic temporal phase modulations −ψt2/2 and ψt2/2 lead to a spectral intensity of the field after modulation I−ψ(ω) and Iψ(ω). The spectral intensity I(ω) and spectral phase φ(ω) can be reconstructed directly from I−ψ(ω) and Iψ(ω) according to:
                              I          ⁡                      (            ω            )                          =                                            I              ave                        ⁡                          (              ω              )                                =                                                                      I                  ψ                                ⁡                                  (                  ω                  )                                            +                                                I                                      -                    ψ                                                  ⁡                                  (                  ω                  )                                                      2                                              (        1        )                                                      ∂                          ∂              ω                                ⁡                      [                                          I                ⁡                                  (                  ω                  )                                            ⁢                              ∂                                  ∂                  ω                                            ⁢                              {                                                      φ                    ⁡                                          (                      ω                      )                                                        +                                                                                    ω                                                  (                          2                          )                                                                    ⁢                                              ω                        2                                                              4                                                  }                                      ]                          =                                            Δ              ⁢                                                          ⁢                              I                ⁡                                  (                  ω                  )                                                                    2              ⁢              ψ                                =                                                                      I                  ψ                                ⁡                                  (                  ω                  )                                            -                                                I                                      -                    ψ                                                  ⁡                                  (                  ω                  )                                                                    2              ⁢              ψ                                                          (        2        )            In Eq. 2, where φ(2) is the second order dispersion of the modulating device. It is noted that the right hand sides of the Eqs. 1 and 2 are an approximation for small modulation (ψ<<1). Those skilled in the art will appreciate that for pulses with an electric field that would not be significantly modified by the dispersion of the modulating device, Eq. 2 can be simplified by using φ(2)=0.
In practice, negative and positive quadratic temporal phase modulations can be sufficiently approximated using a phase modulator driven by a sinusoidal drive signal synchronized with light pulses from an optical source, with an adjustable timing between the pulses and the modulation. FIGS. 1a and 1b respectively represent such timing adjustment for negative and positive quadratic temporal phase modulation.
Using an implementation of simplified chronocyclic tomography based on an optical spectrum analyzer that sequentially measures the optical spectra I−ψ(ω) and Iψ(ω) (FIG. 1c), 2.4-ps pulses from a mode-locked fiber laser were accurately characterized. However, the sensitivity was limited to 1 mW. The slow measurement speed of the scanning optical spectrum analyzer also limited the overall measurement speed. Additionally, since the differential quantity ΔI(ω)=Iψ(ω)−I−ψ(ω) (FIG. 1d) in Eq. 2, is obtained by sequential measurements of the two optical spectra Iψ and I−ψ, the sensitivity of the spectral phase retrieval is limited due to the laser amplitude noise, and thermal and mechanical drifts of the laser and measurement equipment. This is, however, only a particular problem of such sequential implementation of simplified chronocyclic tomography, and the sensitivity would be greatly improved if the differential quantity ΔI(ω) could be measured directly.